Any physical quantity that is proposed in mathematical equations when studying a particular natural phenomenon carries some meaning. The moment of inertia is not an exception to this rule. The physical meaning of this quantity is considered in detail in this article.
Moment of Inertia: Mathematical Formulation
First of all, it should be said that the physical quantity in question is used to describe rotation systems, that is, such movements of an object that are characterized by circular paths around a certain axis or point.
We give the mathematical formula of the moment of inertia for the material point:
I = m * r 2 .
Here m and r are the mass and radius of rotation of the particle (distance to the axis), respectively. Any solid body, no matter how complex it may be, can be mentally divided into material points. Then the formula of the moment of inertia in general will look like:
I = âĢ m r 2 dm.
This expression is always true, and not only for three-dimensional, but also for two-dimensional (one-dimensional) bodies, that is, for planes and rods.
From these formulas it is difficult to understand the meaning of the physical moment of inertia, but an important conclusion can be drawn: it depends on the distribution of mass in the body that is spinning, as well as on the distance to the axis of rotation. Moreover, the dependence on r is sharper than on m (see the sign of the square in the formulas).
Circular motion
It is impossible to understand what the physical meaning of the moment of inertia is if we do not consider the circular motion of bodies. Without going into details, we immediately give two mathematical expressions that describe rotation:
I 1 * Ī 1 = I 2 * Ī 2 ;
M = I * dĪ / dt.
The upper equation is called the law of conservation of the quantity L (angular momentum). It means that no matter what changes occur inside the system (at first there was an inertia moment I 1 , and then it became equal to I 2 ), the product of I by the angular velocity Ī, that is, the angular momentum, will remain unchanged.
The lower expression demonstrates a change in the rotation speed of the system (dĪ / dt) when a certain moment of force M acts on it, which has an external character, that is, is generated by forces that are not related to internal processes in the system under consideration.
In both the upper and lower equations, I is present, and the larger its value, the smaller will be the angular velocity Ī or the angular acceleration dĪ / dt. This is the physical meaning of the moment of inertia of the body: it reflects the ability of the system to maintain its angular velocity. The greater I, the more this ability manifests itself.
Linear momentum analogy
Now we come to the same conclusion that was voiced at the end of the previous paragraph, drawing an analogy between rotational and translational movements in physics. As you know, the latter is described by the following formula:
p = m * v.
This simple expression defines the momentum of the system. Let us compare its shape with that for the angular momentum (see the upper expression in the previous paragraph). We see that the quantities v and Ī have the same meaning: the first characterizes the rate of change of the linear coordinates of the object, the second - the angular coordinates. Since both formulas describe the process of uniform (equiangular) motion, the quantities m and I should also have the same meaning.
Now consider the 2nd law of Newton, which is expressed by the formula:
F = m * a.
Paying attention to the form of writing the lower equality in the previous paragraph, we have a similar situation considered. The moment of force M in its linear representation is the force F, and the linear acceleration a is completely analogous to the angular dĪ / dt. And again we come to the equivalence of mass and moment of inertia.
What is the meaning of mass in classical mechanics? It is a measure of inertia: the larger m, the more difficult it is to move an object from its place, and even more so to give it acceleration. The same can be said about the moment of inertia as applied to the motion of rotation.
The physical meaning of the moment of inertia on a household example
We ask a simple question about how it is easier to twist a metal rod, for example, reinforcement - when the axis of rotation is directed along its length or when across? Of course, it is easier to untwist the rod in the first case, because its moment of inertia for this axis position will be very small (it is zero for a thin rod). Therefore, it is enough to squeeze the object between the palms and with a slight movement bring it into rotation.
By the way, the described fact was experimentally checked by our ancestors back in ancient times, when they learned to make fire. They unwound the stick with huge angular accelerations, which led to the creation of large friction forces and, as a result, to the release of a significant amount of heat.
Auto flywheel - a vivid example of using the large value of the moment of inertia
In conclusion, I would like to give, perhaps, the most important example for modern technology of using the physical meaning of the moment of inertia. Auto flywheel is a solid steel disk with a relatively large radius and mass. These two quantities determine the existence of a significant quantity I characterizing it. The flywheel is designed to "soften" any force on the crankshaft of the car. The impulsive nature of the acting moments of forces from the engine cylinders to the crankshaft is smoothed out and made smooth due to the heavy flywheel.
By the way, the larger the angular momentum, the more energy is in the rotating system (analogy with mass). Engineers want to use this fact, storing the braking energy of the car in the flywheel, in order to subsequently direct it to accelerate the vehicle.